Solve the system of linear equations using matrices. x+y+z=3 2x+3y+2z=7 3x-4y+z=4

Solve the system of linear equations using matrices. x+y+z=3 2x+3y+2z=7 3x-4y+z=4

Question
Matrices
asked 2021-02-13
Solve the system of linear equations using matrices.
x+y+z=3
2x+3y+2z=7
3x-4y+z=4

Answers (1)

2021-02-14
Step 1
Given equation
x+y+z=3
2x+3y+2z=7
3x-4y+z=4
in matrix form
\(A=\begin{bmatrix}1&1&1 \\2&3&2\\3&-4&1 \end{bmatrix}\)
\(B=\begin{bmatrix}3 \\7 \\4 \end{bmatrix}\)
\(X=\begin{bmatrix}x \\y \\z \end{bmatrix}\)
we know that \(A \cdot X=B\)
So, \(X=A^{-1} \cdot B\)
Step 2
Now,
\(X=A^{-1} \cdot B\)
We have to find \(A^{-1}\)
Adjoin the identity matrix onto the right of the original matrix, so that you have A on the left side and the identity matrix on the right side. \(\begin{bmatrix}1&1&1 \\2&3&2\\3&-4&1 \end{bmatrix}=\begin{bmatrix}1&0&0 \\0&1&0\\0&0&1 \end{bmatrix}\)
now we will apply the elementary row method \(R_2-2R_1 \rightarrow R_2\) (multiply 1 row by 2 and subtract it from 2 row), \(R_3-3R_1 \rightarrow R_3\) (multiply 1 row by 3 and subtract it from 3 row)
\(\begin{bmatrix}1&1&1 \\2-2\cdot1&3-2\cdot1&2-2\cdot1\\3-3\cdot1&-4-3\cdot1&1-3\cdot1 \end{bmatrix}=\begin{bmatrix}1&0&0 \\0-2\cdot1&1-2\cdot0&0-2\cdot0\\0-3\cdot1&0-3\cdot0&1-3\cdot0 \end{bmatrix}\\ \begin{bmatrix}1&1&1 \\0&1&0\\0&-7&-2 \end{bmatrix}=\begin{bmatrix}1&0&0 \\-2&1&0\\-3&0&1 \end{bmatrix}\)
\(R_1-1R_2 \rightarrow R_1\) (multiply 2 row by 1 and subtract it from 1 row), \(R_3 + 7 R_2 \rightarrow R_3\) (multiply 2 row by 7 and add it to 3 row)
\(\begin{bmatrix}1&0&1 \\0&1&0\\0&0&-2 \end{bmatrix}=\begin{bmatrix}3&-1&0 \\-2&1&0\\-17&7&1 \end{bmatrix}\)
\(\frac{R_3}{-2} \rightarrow R_3\) (divide the 3 row by -2)
\(\begin{bmatrix}1&0&1 \\0&1&0\\0&0&1 \end{bmatrix}=\begin{bmatrix}3&-1&0 \\-2&1&0\\8.5&-3.5&-0.5 \end{bmatrix}\)
\(R_1-1R_3 \rightarrow R_1\) (multiply 3 row by 1 and subtract it from 1 row)
\(\begin{bmatrix}1&0&0 \\0&1&0\\0&0&1 \end{bmatrix}=\begin{bmatrix}-5.5&2.5&0.5 \\-2&1&0\\8.5&-3.5&-0.5 \end{bmatrix}\)
\(A^{-1}=\begin{bmatrix}-5.5&2.5&0.5 \\-2&1&0\\8.5&-3.5&-0.5 \end{bmatrix}\)
Step 3
Now,
\(X=A^{-1} \cdot B\)
\(X=\begin{pmatrix}-5.5&2.5&0.5 \\-2&1&0\\8.5&-3.5&-0.5 \end{pmatrix}\times \begin{pmatrix}3 \\7 \\4 \end{pmatrix}\)
\(X=\begin{pmatrix}-5.5 \cdot 3&2.5\cdot7&0.5\cdot4 \\-2\cdot3&1\cdot7&0\cdot4\\8.5\cdot3&-3.5\cdot7&-0.5\cdot4 \end{pmatrix}\)
\(X=\begin{pmatrix}3 \\1 \\-1 \end{pmatrix}\)
Answer: x=3,y=1 , z=-1
0

Relevant Questions

asked 2021-02-05

Solve the systems of equations using matrices.
\(4x+5y=8\)
\(3x-4y=3\)
Leave answer in fraction form.
\(4x+y+z=3\)
\(-x+y=-11+2z\)
\(2y+2z=-1-x\)

asked 2021-02-02

Solve the system of equations (Use matrices.):
\(x-2y+z = 16\),
\(2x-y-z = 14\),
\(3x+5y-4z =-10\)

asked 2021-03-29
Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent.
2x-y+z=-3
x-3y=2
x+2y+z=-7
asked 2021-01-25

Solve the system of linear equations using matrices.
\(3x-2y-4=0\)
\(2y=12-x\)

asked 2021-03-15

Use back-substitution to solve the system of linear equations.
\(\begin{cases}x &-y &+5z&=26\\ &\ \ \ y &+2z &=1 \\ & &\ \ \ \ \ z & =6\end{cases}\)
(x,y,z)=()

asked 2021-05-28

Solve by system of equations:
\(5x+3y+z=-8\)
\(x-3y+2z=20\)
\(14x-2y+3z=20\)

asked 2021-02-23
Solve the given set of equations for value of x:
x-3z=-5
2x-y+2z=16
7x-3y-5z=19
asked 2021-03-28
Find all the solutions of the system of equations:
x+2y-z=0, 2x+y+z=0, x-4y+5z=0.
asked 2020-12-25
For the given systems of linear equations, determine the values of \(b_1, b_2, \text{ and } b_3\) necessary for the system to be consistent. (Using matrices)
\(x-y+3z=b_1\)
\(3x-3y+9z=b_2\)
\(-2x+2y-6z=b_3\)
asked 2021-03-10

A system of linear equations is given below.
\(2x+4y=10\)
\(\displaystyle-{\frac{{{1}}}{{{2}}}}{x}+{3}={y}\)
Find the solution to the system of equations.
A. (0, -3)
B. (-6, 0)
C. There are infinite solutions.
D. There are no solutions.

...